In principle the operator \(\widetilde{D}\)  defined by (REF ) makes
sense also in \(L^2\big (\mathbb {R}^2\big )\otimes 2\) , but in this situation, its resolvent is not
compact. This is not an insurmountable problem since it is possible to define spectral triples with the weakest condition that
the elements of the \(C^*\) -algebra are relatively compact with respect to this
Dirac operator. Even though the terminology used in the literature is not
uniform, the situation can be summarized as follows:The distinction
between compact spectral triple vs. locally compact spectral triple used in this paper is borrowed from [1]}.
However, this terminology is not at all standard in the literature.
In our opinion, a better distinction between the two types of spectral triples
should take into account the dichotomy between the discrete and the continuous nature of the spectrum of the Dirac operator. a compact spectral
triple is a spectral triple defined by a Dirac operator with compact resolvent
according to the original definition like [2]}; a locally compact spectral triple is a spectral triple defined by a Dirac operator with respect to which the elements
of the \(C^*\) -algebra are relatively compact [3]} or [1]}. For unital \(C^*\) -algebras every locally compact
spectral triple is automatically compact, in contrast to the non-unital case, where both types of
spectral triples are in general possible. This fact leads to a kind of ambiguity
for the construction of a suitable spectral triple for the magnetic algebra
\({C}_B\) . One possibility is to consider \({C}_B\)  as a noncommutative locally
compact space by endowing this algebra with the geometry defined by the Dirac
operator \(\widetilde{D}\) . This approach has already been explored in the
literature on topological phases of matter in [5]}, [6]}, and
in a more embryonic form in [7]}, [8]}. In particular, the Dirac
operator \(\widetilde{D}\)  allows
a complete description of the QHE and (other topological phases) in the continuum [5]}.
A closely
related approach, already considered in the literature in different contexts [10]}, [11]}, [12]}, consists in considering
the Dirac operator \(\widehat{D}=\sum _{j=1}^2(-\,\mathrm {i}\,\partial _j)\otimes \sigma _j\) 
which is the Fourier transform of \(\widetilde{D}\) . The latter choice also allows
to prove a formula of the type (REF ) by combining the result of [13]} with the observation that \(\big |\widehat{D}\big |^2\)  is proportional to the Laplacian \(\Delta \)  on \(\mathbb {R}^2\) . A completely different approach consists in constructing a compact spectral triple for the magnetic algebra \({C}_B\) . To the best of our knowledge, this attempt
seems to be unexplored in the literature and is the main aim of the
present work.
